<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">L</font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">s</font> be   <a href="struct_0.html#NM7">FinSequence</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>







<b>assume </b><a NAME="E1:60"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">x</font> <a href="polynom8.html#R1" target="_self">is_primitive_root_of_degree</a> <font color="Maroon">m</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp; ( for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> holds <br/><font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a>  <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Olive">k</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> ) )
 ;<br/>

<a NAME="E2:60"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E5:60"><i><font color="Green">E56</font></i></a>, </i>;<br/>
<a NAME="E3:60"/><i><font color="Green">E39</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span> <a href="matrix_3.html#K4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>proof </b></a><div class="add">

<a NAME="E1:60_1"/><i><font color="Green">E42</font></i>: 
(  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>;<br/>
<a NAME="E2:60_1"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>;<br/>
<a NAME="E3:60_1"/><b>then </b>
(  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span> <a href="matrix_3.html#K4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span> <a href="matrix_3.html#K4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> )
 
<b>by </b><i>, <a class="ref" href="matrix_3.html#D4">MATRIX_3:def 4</a></i>;<br/>
<a NAME="E4:60_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span> <a href="matrix_3.html#K4">*</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span> is   <a href="matrix_1.html#NM5">Matrix</a> of <font color="Maroon">m</font>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E5:60"><i><font color="Green">E56</font></i></a>, <a class="ref" href="matrix_1.html#T20">MATRIX_1:20</a></i>;<br/>
<a NAME="E5:60_1"/><b>then </b><i><font color="Green">E51</font></i>: 
 <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span> <a href="matrix_3.html#K4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font></span>)</span></span><a href="mcart_1.html#K12">:]</a></span>
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>;<br/>
<a NAME="E6:60_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E5:60"><i><font color="Green">E56</font></i></a></i>;<br/>
<a NAME="E7:60_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E5:60"><i><font color="Green">E56</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:60_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span> <a href="matrix_3.html#K4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:60"/><i><font color="Green">E55</font></i>: 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2"><b>proof </b></a><div class="add">

<a NAME="E1:60_2"/>
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>;<br/>
<b>hence </b><a NAME="E2:60_2"/>
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>
 <b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:60"/><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span> <a href="matrix_3.html#K4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="fvsum_1.html#K13">"*"</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom8.html#E6:60"><i><font color="Green">E58</font></i></a>, , <a class="ref" href="matrix_3.html#D4">MATRIX_3:def 4</a></i>;<br/>
<a NAME="E6:60"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Olive">k</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:60_3"/><i><font color="Green">E60</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 ;<br/>

<a NAME="E2:60_3"/><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_3_1"><b>proof </b></a><div class="add">

 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span> = 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
<br/>.= 
<font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>
;<br/>
<a NAME="E2:60_3_1"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:60_3_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="txt" href="polynom8.html#E2:60_3_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E4:60_3_1"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="polynom8.html#E6:60_3"><i><font color="Green">E64</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:60_3"/>
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>;<br/>
<a NAME="E4:60_3"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E5:60_3"/><b>then </b><i><font color="Green">E63</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E6:60_3"/><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_3_2"><b>proof </b></a><div class="add">

<a NAME="E1:60_3_2"/><i><font color="Green">E65</font></i>: 
1 <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E2:60_3_2"/>
1 <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="txt" href="polynom8.html#E5:60"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E3:60_3_2"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="polynom8.html#E7:60_3"><i><font color="Green">E66</font></i></a>, <a class="ref" href="xreal_1.html#T129">XREAL_1:129</a></i>;<br/>
<b>hence </b><a NAME="E4:60_3_2"/>
<span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:60_3"/><i><font color="Green">E66</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_3_3"><b>proof </b></a><div class="add">

 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">j</font></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>
<br/>.= 
<font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>
;<br/>
<a NAME="E2:60_3_3"/><b>then </b><i><font color="Green">E67</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">j</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:60_3_3"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E4:60_3_3"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="polynom8.html#E2:60_3_3"><i><font color="Green">E67</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:60_3"/><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_3_4"><b>proof </b></a><div class="add">

<a NAME="E1:60_3_4"/><i><font color="Green">E69</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>;<br/>
<a NAME="E2:60_3_4"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:60_3_4"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom8.html#E8:60_3"><i><font color="Green">E68</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E4:60_3_4"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:60_3"/><i><font color="Green">E70</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_3_5"><b>proof </b></a><div class="add">

<span class="p1">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">j</font> = 
 <a href="polynom8.html#K3" target="_self">pow</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E5:60"><i><font color="Green">E56</font></i></a>, , </i>
<br/>.= 
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , <a class="txt" href="polynom8.html#E1:60"><i><font color="Green">E31</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E2:60_3_5"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E7:60_3"><i><font color="Green">E66</font></i></a>, <a class="txt" href="polynom8.html#E8:60_3"><i><font color="Green">E68</font></i></a>, <a class="txt" href="polynom8.html#E5:60_3"><i><font color="Green">E63</font></i></a>, <a class="txt" href="polynom8.html#E6:60_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="polynom8.html#E5:60"><i><font color="Green">E56</font></i></a>, , </i>
<br/>.= 
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span>
<b>by </b><i>, </i>
;<br/>
<b>hence </b><a NAME="E11:60_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E7:60"/><i><font color="Green">E71</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_4"><b>proof </b></a><div class="add">

 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span> = 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
<br/>.= 
<font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>
;<br/>
<b>then </b> <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_3.html#T8">MATRIX_3:8</a></i>
<br/>.= 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
<br/>.= 
<font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>
;<br/>
<b>hence </b><a NAME="E3:60_4"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:60"/>
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_5"><b>proof </b></a><div class="add">

<a NAME="E1:60_5"/><i><font color="Green">E72</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> holds <br/><span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:60_5_1"/><i><font color="Green">E73</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 ;<br/>

<a NAME="E2:60_5_1"/>
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:60_5_1"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:60_5_1"/><i><font color="Green">E75</font></i>: 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_5_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:60_5_1_1"/><i><font color="Green">E77</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_5_1_1_1"><b>proof </b></a><div class="add">

 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span> = 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
<br/>.= 
<font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>
;<br/>
<a NAME="E2:60_5_1_1_1"/><b>then </b><i><font color="Green">E78</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:60_5_1_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E4:60_5_1_1_1"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:60_5_1_1"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_5_1_1_2"><b>proof </b></a><div class="add">

 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">j</font></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>
<br/>.= 
<font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="matrix_1.html#T25">MATRIX_1:25</a></i>
;<br/>
<a NAME="E2:60_5_1_1_2"/><b>then </b><i><font color="Green">E79</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">j</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:60_5_1_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E4:60_5_1_1_2"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:60_5_1_1"/>
<span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="fvsum_1.html#T73">FVSUM_1:73</a></i>;<br/>


</div><b>end;</b></div>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span> = 
 <a href="polynom8.html#K3" target="_self">pow</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="txt" href="polynom8.html#E5:60"><i><font color="Green">E56</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E6:60_5_1"/>
<span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
 <b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:60_5"/>
 <a href="fvsum_1.html#K11">mlt</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_5_2"><b>proof </b></a><div class="add">

<i><font color="Green">E80</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s</font>
<b>by </b><i><a class="txt" href="polynom8.html#E5:60"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E2:60_5_2"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_5_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:60_5_2_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:60_5_2_2"/><i><font color="Green">E92</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:60_5_2_2"/><b>then </b><i><font color="Green">E93</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:60_5_2_2"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s</font>
 
<b>by </b><i>, , <a class="txt" href="polynom8.html#E5:60"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E6:60_5_2_2"/>
<span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:60_5_2"/>
 <a href="fvsum_1.html#K11">mlt</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:60_5"/>
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K11">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p3">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">s</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:60"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <font color="Maroon">x</font>,<font color="Maroon">m</font></span>)</span> <a href="matrix_3.html#K4">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#NK8" target="_self">VM</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="vectsp_1.html#K4">"</a> </span>)</span>,<font color="Maroon">m</font></span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="ref" href="fvsum_1.html#D10">FVSUM_1:def 10</a></i>;<br/>


</div>
